### Functions: Fractional functions

### Power functions with negative exponents

Power function with negative exponent

A **power function** with a negative integer exponent has the form \[f(x)=\blue{a}x^{-\orange{n}}\]

in which #\orange{n}# is a postive integer.

We can also write this function as \[f(x)=\frac{\blue{a}}{x^{\orange{n}}}\]

The graph of a power function with a negative integer exponent, moves through the point #\rv{1,\blue{a}}#, has a vertical asymptote at #x=0# and horizontal asymptote in the line #y=0#.

If #\orange{n}# is even, the function is symmetrical across the #y#-axis. If #\orange{n}# is odd, the function has the point #\rv{0,0}# as the point of symmetry.

GeoGebra Negative powerfunction

Take a look at the graph of a power function with negative exponent, which is a function of the form #f(x)=\frac{a}{x^n}#.

What do we know about the values of #n# and #a#?

What do we know about the values of #n# and #a#?

The value of #n# is: odd

The value of #a# is: negative

The graph is symmetrical across the point #\rv{0,0}#, hence, the value of #n# is odd.

The #y#-value is negative if the value of #x# is positive, hence, the value of #a# is negative.

The value of #a# is: negative

The graph is symmetrical across the point #\rv{0,0}#, hence, the value of #n# is odd.

The #y#-value is negative if the value of #x# is positive, hence, the value of #a# is negative.

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